Subgroups — #72216539

Order 2048.

499 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72216243	C₂	1	1	yes	maximal, characteristic
2	#72216215	C₂	1	1	yes	maximal, characteristic
3	#72216178	C₂	1	1	yes	maximal, characteristic
4	#151981	—	2	1	no	—
5	C₂×#7459	C₄	1	2	yes	—
6	#129090	C₄	1	1	yes	characteristic
7	#129088	C₄	1	1	yes	characteristic
8	#129051	—	2	1	no	—
9	#126522	—	2	1	no	—
10	C₈×#346	C₂²	1	1	yes	characteristic
11	C₂×#7053	—	2	1	no	—
12	C₂×#7053	C₄	1	2	yes	—
13	#94019	—	2	1	no	—
14	#93264	—	2	1	no	—
15	C₂×#1422	—	2	2	no	—
16	C₂×#1398	—	4	1	no	—
17	C₄×#364	—	2	1	no	—
18	C₄×#346	—	2	2	no	—
19	C₄×#346	C₂×C₄	1	1	yes	characteristic
20	C₄×#346	D₄	1	1	yes	characteristic
21	C₂×#1369	—	2	1	no	—
22	#11999	—	2	1	no	—
23	#11405	—	4	1	no	—
24	C₂×#1312	—	2	2	no	—
25	C₂×#1312	—	2	2	no	—
26	C₂×#1311	C₂×C₄	1	1	yes	characteristic
27	C₂×#1311	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₆	Q₈	1	1	yes	characteristic
30	#8266	—	2	1	no	—
31	C₄×C₈²	D₄	1	1	yes	characteristic
32	#7468	—	4	1	no	—
33	#7459	—	2	2	no	—
34	#7459	C₈	1	4	yes	complemented
35	#7161	C₈	1	2	yes	—
36	#7157	C₈	1	2	yes	—
37	#7140	—	4	1	no	—
38	#7053	—	2	4	no	—
39	#7053	—	2	2	no	—
40	#7053	C₈	1	4	yes	complemented
41	C₂×M₆₄	—	4	2	no	—
42	C₂×#364	—	2	2	no	—
43	C₄×M₃₂	—	4	1	no	—
44	C₂×#346	—	4	4	no	—
45	C₂×#346	C₂×C₈	1	1	yes	characteristic
46	C₂×#346	C₂×C₈	1	1	yes	characteristic
47	C₂×#346	M₁₆	1	1	yes	characteristic
48	C₂×#346	M₁₆	1	1	yes	characteristic
49	C₂×#346	D₈	1	1	yes	characteristic
50	C₂×#346	SD₁₆	1	1	yes	characteristic
51	C₂×#346	SD₁₆	1	1	yes	characteristic
52	C₂×#346	Q₁₆	1	1	yes	characteristic
53	C₂×C₄×C₁₆	—	2	4	no	—
54	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
55	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₈	—	2	1	no	—
59	C₄²×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	#1564	—	2	2	no	—
61	#1450	—	4	1	no	—
62	C₂×C₈²	—	2	1	no	—
63	C₂×C₈²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
65	#1422	—	2	4	no	—
66	#1422	—	4	2	no	—
67	#1398	—	8	2	no	—
68	#1376	—	4	1	no	—
69	#1369	—	4	2	no	—
70	#1368	—	4	2	no	—
71	#1313	—	8	2	no	—
72	#1312	—	2	4	no	—
73	#1312	—	2	4	no	—
74	#1312	—	4	4	no	—
75	#1311	—	2	2	no	—
76	#1311	C₂×C₈	1	2	yes	—
77	#1311	M₁₆	1	2	yes	—
78	C₈×C₁₆	—	2	2	no	—
79	C₈×C₁₆	C₂×C₈	1	2	yes	—
80	C₈×C₁₆	M₁₆	1	2	yes	—
81	#1276	—	4	1	no	—
82	C₂×M₃₂	—	4	2	no	—
83	C₂²×C₁₆	—	4	4	no	—
84	C₂²×C₁₆	C₄×C₈	1	1	yes	characteristic
85	C₂²×C₁₆	#97	1	1	yes	characteristic
86	C₂²×C₁₆	#102	1	1	yes	characteristic
87	C₂²×C₁₆	#103	1	1	yes	characteristic
88	C₂²×C₁₆	C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₁₆	C₄⋊C₈	1	1	yes	characteristic
90	C₂²×C₁₆	#106	1	1	yes	characteristic
91	C₂²×C₁₆	#107	1	1	yes	characteristic
92	#431	—	4	2	no	—
93	C₂×C₄×C₈	—	2	1	no	—
94	C₂×C₄×C₈	—	2	1	no	—
95	C₂×C₄×C₈	—	2	4	no	—
96	C₂×C₄×C₈	#95	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₈	#102	1	1	yes	characteristic
100	C₂×C₄×C₈	#103	1	1	yes	characteristic
101	C₂×C₄×C₈	#104	1	1	yes	characteristic
102	C₄³	#104	1	1	yes	characteristic
103	M₆₄	—	4	4	no	—
104	M₆₄	—	8	2	no	—
105	C₂×C₃₂	—	8	2	no	—
106	#364	—	4	2	no	—
107	#364	—	4	2	no	—
108	#346	—	2	2	no	—
109	#346	—	2	2	no	—
110	#346	—	2	2	no	—
111	#346	—	2	1	no	—
112	#346	—	2	1	no	—
113	#346	—	4	1	no	—
114	#346	—	4	1	no	—
115	#346	—	4	1	no	—
116	#346	—	4	1	no	—
117	#346	—	8	1	no	—
118	#346	—	8	8	no	—
119	C₄×C₁₆	—	2	8	no	—
120	C₄×C₁₆	—	2	1	no	—
121	C₄×C₁₆	—	2	1	no	—
122	C₄×C₁₆	—	4	4	no	—
123	C₄×C₁₆	—	8	1	no	—
124	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
125	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
126	C₄×C₁₆	#97	1	1	yes	characteristic
127	C₄×C₁₆	#97	1	1	yes	characteristic
128	C₄×C₁₆	C₂²⋊C₈	1	2	yes	—
129	C₄×C₁₆	#104	1	2	yes	—
130	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
131	C₄×C₁₆	#108	1	2	yes	—
132	#335	—	4	2	no	—
133	#334	—	4	2	no	—
134	#322	—	8	1	no	—
135	C₈²	—	2	4	no	—
136	C₈²	—	2	1	no	—
137	C₈²	—	2	1	no	—
138	C₈²	—	8	1	no	—
139	C₈²	C₂²⋊C₈	1	2	yes	—
140	C₈²	C₄⋊C₈	1	2	yes	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	D₄⋊C₈	1	1	yes	characteristic
144	C₂²×C₈	Q₈⋊C₈	1	1	yes	characteristic
145	C₂²×C₈	#336	1	1	yes	characteristic
146	C₂²×C₈	#340	1	1	yes	characteristic
147	C₂×C₄²	—	2	2	no	—
148	C₂×C₄²	D₄⋊C₈	1	1	yes	characteristic
149	C₂×C₄²	Q₈⋊C₈	1	1	yes	characteristic
150	C₂×C₄²	#339	1	1	yes	characteristic
151	M₃₂	—	8	2	no	—
152	M₃₂	—	8	2	no	—
153	C₂×C₁₆	—	2	1	no	—
154	C₂×C₁₆	—	2	1	no	—
155	C₂×C₁₆	—	2	1	no	—
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	2	1	no	—
160	C₂×C₁₆	—	2	1	no	—
161	C₂×C₁₆	—	2	1	no	—
162	C₂×C₁₆	—	2	1	no	—
163	C₂×C₁₆	—	2	1	no	—
164	C₂×C₁₆	—	2	1	no	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	1	no	—
167	C₂×C₁₆	—	4	8	no	—
168	C₂×C₁₆	—	8	8	no	—
169	C₂×C₁₆	—	8	2	no	—
170	C₂×C₁₆	C₈²	1	1	yes	characteristic
171	C₂×C₁₆	#322	1	1	yes	characteristic
172	C₂×C₁₆	#322	1	1	yes	characteristic
173	C₂×C₁₆	#322	1	1	yes	characteristic
174	C₂×C₁₆	D₄⋊C₈	1	2	yes	—
175	C₂×C₁₆	Q₈⋊C₈	1	2	yes	—
176	C₂×C₁₆	#334	1	2	yes	—
177	C₂×C₁₆	#335	1	2	yes	—
178	C₂×C₁₆	#347	1	2	yes	—
179	C₂×C₁₆	#347	1	2	yes	—
180	C₂×C₁₆	#360	1	2	yes	—
181	C₂×C₁₆	#361	1	1	yes	characteristic
182	C₂×C₁₆	#362	1	1	yes	characteristic
183	C₄⋊C₈	—	8	2	no	—
184	C₂²⋊C₈	—	8	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	4	8	no	—
199	C₄×C₈	—	8	2	no	—
200	C₄×C₈	#336	1	1	yes	characteristic
201	C₄×C₈	#336	1	1	yes	characteristic
202	C₄×C₈	#339	1	1	yes	characteristic
203	C₄×C₈	#341	1	1	yes	characteristic
204	C₃₂	—	8	4	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	#1279	1	1	yes	characteristic
208	C₂²×C₄	#1336	1	2	yes	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	4	no	—
218	C₂×C₈	—	16	2	no	—
219	C₂×C₈	#1273	1	1	yes	characteristic
220	C₂×C₈	#1277	1	1	yes	characteristic
221	C₂×C₈	#1278	1	1	yes	characteristic
222	C₂×C₈	#1279	1	1	yes	characteristic
223	C₂×C₈	#1336	1	2	yes	—
224	C₂×C₈	#1375	1	1	yes	characteristic
225	C₂×C₈	#1375	1	1	yes	characteristic
226	C₂×C₈	#1378	1	1	yes	characteristic
227	C₂×C₈	#1378	1	1	yes	characteristic
228	C₂×C₈	#1386	1	1	yes	characteristic
229	C₂×C₈	#1387	1	1	yes	characteristic
230	C₂×C₈	#1388	1	1	yes	characteristic
231	C₂×C₈	#1389	1	1	yes	characteristic
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	4	no	—
238	C₄²	#1276	1	1	yes	characteristic
239	C₄²	#1278	1	1	yes	characteristic
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	1	no	—
242	C₁₆	—	2	1	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	4	1	no	—
246	C₁₆	—	4	1	no	—
247	C₁₆	—	8	8	no	—
248	C₁₆	—	16	2	no	—
249	C₁₆	#1313	1	2	yes	—
250	C₁₆	#1313	1	2	yes	—
251	C₁₆	#1333	1	2	yes	—
252	C₁₆	#1334	1	2	yes	—
253	C₂³	#7182	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	#7015	1	1	yes	characteristic
262	C₂×C₄	#7168	1	1	yes	characteristic
263	C₂×C₄	#7168	1	1	yes	characteristic
264	C₂×C₄	#7175	1	1	yes	characteristic
265	C₂×C₄	#7176	1	1	yes	characteristic
266	C₂×C₄	#7182	1	1	yes	characteristic
267	C₈	—	2	8	no	—
268	C₈	—	2	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	1	no	—
271	C₈	—	32	2	no	—
272	C₈	#7137	1	1	yes	characteristic
273	C₈	#7140	1	1	yes	characteristic
274	C₈	#7158	1	1	yes	characteristic
275	C₈	#7159	1	1	yes	characteristic
276	C₈	#7164	1	1	yes	characteristic
277	C₈	#7164	1	1	yes	characteristic
278	C₈	#7177	1	1	yes	characteristic
279	C₈	#7178	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	#92943	1	1	yes	characteristic
282	C₂²	#92944	1	1	yes	characteristic
283	C₂²	#93266	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	1	no	—
288	C₄	#92943	1	1	yes	characteristic
289	C₄	#92944	1	1	yes	characteristic
290	C₄	#93261	1	1	yes	characteristic
291	C₄	#93270	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	#72214472	1	2	yes	—
295	C₂	#72216538	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#7175
commutator	C₂×C₁₆	C₈²
Frattini, cosocle	C₈×#346	C₂²
socle	C₂²	#92943
prime radical	C₂³	#7182
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
34. #7459	C₈	semidirect
40. #7053	C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#72216243	2
#72216215	2
#72216178	2